Regularity of quotients of Drinfeld modular schemes

Satoshi Kondo and Seidai Yasuda

Vol. 304 (2020), No. 2, 481–503

DOI: 10.2140/pjm.2020.304.481

Abstract

Let $A$ be the coordinate ring of a projective smooth curve over a finite field minus a closed point. For a nontrivial ideal $I\subset A$, Drinfeld defined the notion of structure of level $I$ on a Drinfeld module.

We extend this to that of level $N$, where $N$ is a finitely generated torsion $A$-module. The case where $N={\left(I^{-1}∕A\right)}^d$, where $d$ is the rank of the Drinfeld module, coincides with the structure of level $I$. The moduli functor is representable by a regular affine scheme.

The automorphism group ${Aut}_A\left(N\right)$ acts on the moduli space. Our theorem gives a class of subgroups for which the quotient of the moduli scheme is regular. Examples include generalizations of ${\Gamma }_0$ and of ${\Gamma }_1$. We also show that parabolic subgroups appearing in the definition of Hecke correspondences are such subgroups.

Keywords

Drinfeld modular schemes, Drinfeld level structures, regularity

Mathematical Subject Classification 2010

Primary: 11F32

Secondary: 14G35

Milestones

Received: 21 August 2017

Revised: 10 August 2019

Accepted: 11 August 2019

Published: 12 February 2020